Matrices and eigen value problems

Let A and B be nxn matrices over reals. Show that I - BA is invertible if I - AB is invertible. Deduce that AB and BA have the same eigen values

2. Relevant equations

det(AB) = det(A).det(B)

3. The attempt at a solution

given: (I-AB) is invertible

-> det(I-AB) is not equal to 0

i.e. (-1)^n times det(AB-I) is not equal to 0
i.e. (-1)^n times det(AB)-det(I) is not equal to 0
i.e. (-1)^n times det(A).det(B)-det(I) is not equal to 0
i.e. (-1)^n times det(B).det(A)-det(I) is not equal to 0
i.e. (-1)^n times det(BA)-det(I) is not equal to 0
i.e. (-1)^n times det(BA-I) is not equal to 0

If those extra terms vanish, then [itex] \det (A+B) =\det A +\det B[/itex]. But in general, it's not true. It would be nice if it were though. Remember it's ONLY linear in the rows and columns, not the entire matrix itself. This is also the reason that if A is nxn, then

If those extra terms vanish, then [itex] \det (A+B) =\det A +\det B[/itex]. But in general, it's not true. It would be nice if it were though. Remember it's ONLY linear in the rows and columns, not the entire matrix itself. This is also the reason that if A is nxn, then

[tex] \det (cA) = c^n \det A.[/tex]

thanks a lot.. any idea then how to solve the first part of the problem.. i am stuck...

For future reference, start a new thread when you have a new question. You'll get a lot more traffic and lots more help.

EDIT: I wrote a big long post, but then forgot to ask the most important question first! Can you show me what you've tried so far?

well what i have tried so far has been the wrong approach as you have pointed out.. thanks for that ..... so to be honest ... right now i have nothing to show .. as i didn't understand how really B(I−AB)^−1.A+I is the inverse to the matrix i am looking for??????

There are a variety of ways to show that a matrix is invertible: show it's determinant is different from zero, show that it has full rank, show that the equation Ax=0 has only the trivial solution, etc. etc. However, we're going to construct a matrix that will send I-BA to the identity, and since inverses are unique, this must be the inverse we're after. Also note that--in the case of matrices at least--showing a matix is a left- or right-inverse is equivalent to showing that it is THE two-sided inverse.

So we need to show that

[tex] (I-BA)(B(I-AB)^{-1}A+I) = I.[/tex]

That matrix, our candidate for the inverse, exists by hypothesis, i.e., it contains the inverse of I-AB which we are allowed to assume exists. So, therefore, the first step is to expand that multiplication:

There are a variety of ways to show that a matrix is invertible: show it's determinant is different from zero, show that it has full rank, show that the equation Ax=0 has only the trivial solution, etc. etc. However, we're going to construct a matrix that will send I-BA to the identity, and since inverses are unique, this must be the inverse we're after. Also note that--in the case of matrices at least--showing a matix is a left- or right-inverse is equivalent to showing that it is THE two-sided inverse.

So we need to show that

[tex] (I-BA)(B(I-AB)^{-1}A+I) = I.[/tex]

That matrix, our candidate for the inverse, exists by hypothesis, i.e., it contains the inverse of I-AB which we are allowed to assume exists. So, therefore, the first step is to expand that multiplication:

It's hard to know what exactly to recommend when I don't know your background.

There is a subforum over in Academic Guidance called Science Book Discussion where they will recommend books. Maybe you should post over there. Or, I'm sure there will probably be some results that pop up if you use the search feature. And, take a look https://www.physicsforums.com/blog.php?b=3206" [Broken].

I didn't learn my linear algebra and matrix theory all from one source, it was kind of a piecemeal education that I picked up as I went along. I hated my intro linear algebra text and course, so perhaps that's why. Now it's one of my favorite things to talk about.

It's hard to know what exactly to recommend when I don't know your background.

There is a subforum over in Academic Guidance called Science Book Discussion where they will recommend books. Maybe you should post over there. Or, I'm sure there will probably be some results that pop up if you use the search feature. And, take a look https://www.physicsforums.com/blog.php?b=3206" [Broken].

I didn't learn my linear algebra and matrix theory all from one source, it was kind of a piecemeal education that I picked up as I went along. I hated my intro linear algebra text and course, so perhaps that's why. Now it's one of my favorite things to talk about.